Adaptive Vs

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)

ADAPTIVE VS. HYBRID ITERATIVE MIMO RECEIVERS BASED ON MMSE LINEAR AND SOFT-SIC DETECTION Ernesto Zimmermann and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, TU Dresden, D-01062 Dresden, Germany A BSTRACT In this paper, we propose the use of a combination of linear MMSE and soft successive interference cancellation based detectors for the application in (Turbo-)MIMO receivers. In an adaptive setup we switch between the two detector types based on the quality of the received signal, while in a hybrid setup, we use a linear MMSE in the first and a SoftSIC in all subsequent iterations. We show that for a number of scenarios, both techniques show performance close to that of more sophisticated detection techniques, at a fraction of the complexity.

I.

I NTRODUCTION

As engineers strive to satisfy the demand for ever higher data rates in future wireless systems, they are faced with a serious challenge: radio frequency spectrum is often limited by regulation and other factors. Multiple-input multiple-output (MIMO) systems are a promising approach to using this scarce resource as efficiently as possible by multiplexing several data streams into the same time-frequency bin [1]. A big challenge in this context is the correct separation of the transmitted signals at the receiver. Recently, the field of iterative “Turbo” MIMO detection based on the serial concatenation of an inner MIMO detector and an outer channel decoder has received a lot of attention. Sphere detection [2] for example enables approaching the MIMO channel capacity while avoiding the prohibitive complexity of a full APP detector. However, it is often forgotten that simple linear MMSE equalization performs remarkably close to maximum likelihood (ML) detection in non-iterative MIMO receiver setups. On the other hand, soft successive interference cancellation [3] (SoftSIC) based detectors show very good performance in iterative setups – but tend to be limited by error propagation when used with higher order modulation schemes. In this contribution we show how a combination of the two techniques can be used to achieve excellent detection performance at very low complexity. We compare two different approaches: a hybrid detector, where the linear MMSE is used in the first and the SoftSIC in all subsequent iterations, and an adaptive detector that flexibly switches between the two detector types, steered by available soft information. The remainder of this document is structured as follows: Section II. introduces the system model and principles of iterative MIMO detection. In Section III. we describe the investigated receiver algorithms. Performance results for a number of relevant channel models are given in Section IV. before we summarize our findings in Section V.. c 1-4244-0330-8/06/$20.00
2006 IEEE

S YSTEM M ODEL

II.

A. MIMO Model We consider a MIMO system with M transmit and N receive antennas, as depicted in Figure 1. Let u be a vector of i.i.d. information bits which are encoded using the outer channel code, and interleaved. The resulting code bit stream is partitioned into blocks c of M · L bits, where L denotes the number of bits per symbol (allowing to distinguish between Q = 2L different constellation points). Each block c = (c1 , · · · , cM )T consists of M binary vectors cm , m = 1, · · · , M of L bits and is mapped onto a vector symbol x = (x1 , · · · , xM )T whose components are taken from some complex constellation C (e.g. 16-QAM) using the mapping function xm = map(cm ) (e.g., Gray mapping). Binary Source

u

Outer Encoder

e

Rate R

x ... H ...

Interleaver

AWGN Hard Decision Binary Sink

Constellation Mapper

c

n y

SISO Decoder

LA,Dec

-1

LE,Dec

LE,Det

MIMO Detector

LA,Det

Figure 1: Transmission model with outer channel encoder, MIMO channel and iterative receiver (soft-input soft-output detector and decoder). We consider transmission over a frequency flat fading MIMO channel. In the equivalent base-band model, the received signal yt at time index t is thus given by yt = Ht xt + nt

(1)

where Ht ∈ CN ×M is the channel transfer matrix assumed to be perfectly known at the receiver. The entries of Ht are realizations of zero mean i.i.d. complex Gaussian random processes of variance 1 (i.e., each subchannel is passive). We normalize the average transmit energy such that N ×1 represents the E{xt xH t } = Es /M I. The vector nt ∈ C receiver noise and its components are zero mean i.i.d. complex Gaussian random variables with variance N0 /2 per real dimension: E{nt nH t } = N0 I. The signal-to-noise ratio (SNR) at each receive antenna is hence given by SNR = Es /N0 . We also define σ 2 = M N0 /Es . B. Iterative Detection and Decoding We consider the serial concatenation of an inner MIMO detector and an outer channel decoder. Both entities are able to

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)

accept and generate soft information, which are exchanged between them during the iterative reception process. The detector uses the received signal, the channel state information and the a-priori information provided by the decoder to generate extrinsic information on the received bits. The channel decoder uses the correlation between different code bits introduced by the encoder (i.e., the code’s structure) to generate extrinsic information about the information bits as well as the code bits. The latter information is interleaved and fed back to the detector. Information is exchanged between detector and decoder using the so called log-likelihood ratios (LLRs) for the code bits ci (dropping time index t for ease of notation): L(ci |y)

P [ci = +1|y] P [ci = −1|y]


:=

ln

=

ln


x(c)∈Xci =+1 x(c)∈Xci =−1

p (y|x(c)) · P [c] p (y|x(c)) · P [c]

.

(2)

where the second line follows from Bayes’ theorem and the assumption of statistically independent bits. Xci =±1 denotes the set of 2M ·L−1 symbols x for which ci = ±1. The second term in (2) represents the a-priori knowledge from the outer channel decoder. The conditioned probability densities in (2) are given by the complex Gaussian distribution:   1 1 2 exp − y − Hx . (3) p(y|x) = (πN0 )N N0 For the LLR computation, the constant scaling factor cancels out and can thus be omitted. To evaluate the numerator and denominator of (2) at low complexity, it is useful to apply the so called “maxLog” approximation [4]:   2  − y − Hx max + ln P [ci ] (4) L(ci |y) ≈ x∈Xci =+1 N0 i   2  − y − Hx − max + ln P [ci ] . x∈Xci =−1 N0 i Evaluating the two max-operations in equation (4) by a bruteforce approach (APP detection) is well known to require an effort growing exponentially in the number of transmitted bits per vector symbol, that is, the achieved (raw) spectral efficiency. However, there exist a number of low complexity algorithms that show very good performance at only a small fraction of the full APP complexity. We will discuss some example techniques and their properties in the following section. For channel coding, we consider the parallel concatenation of two recursive systematic convolutional codes – a classical “Turbo code”. The decoder employs two instances of the BCJR [5] algorithm that exchange extrinsic information in a number of (internal) decoder iterations and afterwards feed back extrinsic information to the MIMO detector. III. A.

L OW C OMPLEXITY MIMO D ETECTION

MMSE Linear Detection (MMSE-LD)

Linear equalization is the straightforward (and lowest complexity) detection scheme for MIMO. By inverting the channel and

thus suppressing the interference among the layers, it turns the MIMO detection problem into a set of M parallel SISO detection problems: y ˜

= Gy = GHx + Gn = Ψx + n ˜

(5)

where G is the linear filtering matrix. The order of complexity of demapping (neglecting any preprocessing) is substantially reduced from O(2M ·L ) to O(2L ), at the expense of (potentially severe) noise enhancement and a reduction of the spatial diversity order to 1. Ψ models the residual interference among the layers, while n ˜ models the noise at the equalizer output, which is in general no longer white. For the zero forcing (ZF) case, GZF := (HH H)−1 HH and thus Ψ = I – the interference among layers is completely removed. For the important case of MMSE detection, which aims at limiting noise enhancement, the filtering matrix is defined as: GMMSE := (HH H + σ 2 I)−1 HH

(6)

and Ψ = I – some interference among the layers remains. Also, in general, dg (Ψ) = I1 – the concatenation of the channel and the linear filter has no longer unit gain. In order to remove this bias, the filtering matrix may be modified as ˜ MMSE := GMMSE S, where we define the scaling matrix G S = diag(1/ψ1,1 , . . . , ψm,m ). The post-equalization SINR γm on each of the layers m is given by (e.g. [6]): γm,LD

=

1 1   σ 2 (HH H + σ 2 I)−1

− 1, m,m

and the LLRs for each layer readily follow as:  ym ) ≈ L(ci |˜

max

x∈Cci =+1

− max

x∈Cci =−1

2

−γm ˜ ym − x +

... .



 ln P [ci ]

i

(7)

The full decoupling of the layers has another drawback: there is little or no gain from using iterative techniques together with linear detectors and Gray mapping, as the EXIT characteristic of the linear detector is almost horizontal (cf. Figure 2). What nevertheless renders simple linear detection attractive is that the noise enhancement due to the linear filtering can be characterized very precisely. Thus, the quality of the soft output (magnitudes of the LLRs) is relatively high. The difference in the quality of the hard output relative to optimal ML/MAP detection on the other hand depends strongly on the operating regime. For low rate channel coding (code rate 1/2 and below) and high diversity regimes, the target operating regime is at an uncoded BER above 10% (remember that the Shannon bound for rate 1/2 coding on the BSC is at a crossover rate of roughly 15%). In such a setup, linear detection is a promising alternative to more advanced detection techniques. 1 We define dg(·) = diag−1 (diag(·)), i.e., the result is a diagonal matrix containing only the diagonal entries of the argument.

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)

B.

The SoftSIC post-equalization SINR now follows as:

MMSE-SoftSIC Detection

The basic idea of soft successive interference cancellation is to subtract soft estimates of previously detected signal from the received signal, and use the variance of these soft estimates to assess the expected residual noise after the cancellation step [3]. In order to perform the interference cancellation, the channel matrix H is decomposed to obtain a matrix of upper triangular form. It is obviously advantageous to detect the strongest antenna first in order to minimize the effects of error propagation. We use the MMSE based sorted QR decomposition [7] for this purpose. Employing the system model from [8]   H y H = y= , (8) σI 0 the MMSE extended channel matrix H is decomposed into a matrix R ∈ CM ×M of upper-triangular structure and a matrix Q ∈ C(N +M )×M of orthogonal columns:   Q1 Q1 R H = QR= R= . (9) Q2 Q2 R Obviously, Q is non-square and therefore no longer unitary/orthonormal (in contrast to the ZF-SQRD case). MultiH plying the received signal with Q , we obtain [6]: y ˜= =

H

H H H H Q y = Q1 Hx + Q1 n = Rx − Q2 σx + Q1 n  H   H   H  H x − Q2 − dg Q2 R − σdg Q2 nx + Q1 n

˜ +n = Rx ˜

(10)

˜ can be employed for unbiased MMSE based SIC where R detection. The second term in the second line of (10) models the residual interference among layers (note that

˜m := E{˜ E{nx nH } = N I). We further define N n n ˜ H } m,m 0 x as the total variance of the “channel noise” on layer m, including the interference among layers as well as the receiver noise. Note that using the unbiased MMSE solution for both, linear detection and SoftSIC is essential for the design of the adaptive detector (cf. Section C.). Exploiting the upper triangular struc˜ the L-value from the detector is now given by: ture of R,

γm,SIC

rm,m |2 rm,m |2 N0 Es /M |˜ 1 |˜ = 2 (13) ˜m + NIC,m ˜m + NIC,m σ N N

=

Here, NIC,m denotes the residual interference from the cancellation step on layer m and can be calculated as [3]: NIC,m =

M 

|˜ rm,j |2 · Var[xj ].

(14)

j=m+1

with the
variance of the soft symbols given by Var[xj ] = (xq − x ¯j ) · P [xq ], where the symbol probabilities required for soft symbol computation can be reused. There are two major drawbacks of SoftSIC detection, and both have the highest impact when no or little extrinsic information from the decoder is available (i.e., in the first iteration). Firstly, error propagation substantially degrades the SoftSIC detector’s performance. Secondly, the calculation of the soft symbols and the cancellation noise is computationally expensive, as almost all possible constellation points have to be considered, unless a priori knowledge is available. In this case, substantial reductions in complexity can be achieved [9]. C.

An Adaptive MMSE-SoftSIC/Linear Detector

Our aim is to design a detection algorithm that alleviates the error propagation problems of the MMSE-SoftSIC detector, when no or only little a priori knowledge is available. Evidently, the new detector should perform at least as good as linear MMSE detection. There exists in fact a close connection between successive interference cancellation and linear detection, that we will exploit in the design of the adaptive detector: remember the interference reduced signal used as a basis for the LLR calculation in the SoftSIC detector in (11):
ˆj y˜m − rm,j x . (15) x ˜m = rm,m

If we use the unquantized signal x ˜j from the previous layers, ˆj , the instead of the soft symbol xj for the signal estimate x detector output will be equivalent to that of the linear MMSE detector. We can use this equivalence very conveniently to con M  2 struct a detector that flexibly switches between SoftSIC and lin r˜m,j  y˜m  L(ci |˜ ym ) ≈ max − x ¯j − xm  ear detection. We first calculate γm,SIC using the soft symbols − γm,SIC  x∈Cci =+1 r˜m,m j=m+1 r˜m,m xj from previously detected layers. If this SINR is superior to



 the SINR γm,LD at the output of the linear detector, we directly + ln P [ci ] − max ··· , use (11) to calculate the LLRs in this layer. In the contrary case x∈Cci =−1 ˜j in (11) to obtain the linear MMSE (11) we replace xj replaced by x filter output. In order to avoid having to evaluate equation (16) where the soft symbols x ¯j are defined as: to determine γm,LD , the following derivations can be used: x ¯j

:=

Q  q=1

xq · P [xq ] =

Q  q=1

xq ·

L 

P [cj,l ].

(12)

l=1

with xq ∈ C. Note that we use the L-value including the channel state information and the a priori knowledge for computing the bit/symbol probabilities and to generate a high quality estimate of the transmitted symbol.

HH H + σ 2 I

−1

=



−1  H −1 1 H H H H = R R = 2 Q2 Q2 σ (16) −1

= 1/σ Q2 [7]. The effort rewhere used the fact that R quired for this new adaptive detector (we will use the term SoftSIC/LD in the following) is hence largely the same as for the

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)

D.

EXIT Chart Analysis

Figure 2 shows the EXIT characteristics of MMSE-LD, MMSE-SoftSIC and MMSE-SoftSIC/LD for a 4x4 system using 16-QAM (solid lines, SNR= 10.5dB) and 256-QAM (dashed lines, SNR= 20.5dB) transmission over the i.i.d. fast fading channel from [2] – the 802.11n channel model E shows similar characteristics. For 16-QAM transmission, the linear detector performs only slightly better than the SoftSIC when no a priori knowledge is available (LA,Det =0). However, for 256-QAM, the error propagation in the SoftSIC becomes a real issue and there is a significant performance degradation w.r.t. the linear MMSE detector (this is confirmed by the simulation results in the following section).

100MHz system bandwidth, 596 data subcarriers, 6.4 µs OFDM symbol length, 0.8 µs cyclic prefix), using 4×4 MIMO and 16-QAM, 64-QAM and 256-QAM transmission. For channel coding we used a rate 1/2 parallel concatenated (7R , 5) Turbo code. The decoder performed 8 internal decoding iterations and the interleaver size is 14304 bits for 16-/64-QAM and 9536 bits for 256-QAM (3,2 and 1 OFDM symbol, respectively). We used IEEE 802.11n channel models B and E as examples for a low and high (spatial and frequency) diversity regime, respectively. 0

10

16−QAM

−1

10

1 −2

10

0.9 0.8

IE,Det; IA,Dec

0.6

x ... MMSE−LD + ... MMSE−SoftSIC o ... MMSE−SoftSIC/LD v ... Sphere

12

0.7

256−QAM

64−QAM

FER

SoftSIC detector. The only required overhead is the storage of the unquantized signals x ˜j , the calculation of the diagonal elements of the matrix from (16), which is not data dependent and can therefore be done as a part of the preprocessing; and an additional comparison before the evaluation of (11).

14

16

18

20

22 24 SNR [dB]

26

28

30

32

MMSE−SoftSIC MMSE−SoftSIC/LD

0.5

Figure 3: Performance results for single-shot equalization, IEEE 802.11n B channel model

MMSE−LD

0.4 Rate 0.5 PCCC,

0.3 (based on mem. 2 CC, 8 logMAP its.)

0.2 0.1 0 0

0.1

0.2

0.3

0.4 I

0.5 ;I

0.6

0.7

0.8

0.9

1

A,Det E,Dec

Figure 2: Extrinsic information transfer chart for a linear MMSE (near horizontal curves), a SoftSIC (upper curves) and the SoftSIC/LD detector (circle markers), for 16-QAM (dashes curves) and 256-QAM (solid curves). For high a priori knowledge, the SoftSIC significantly outperforms the linear detector, as the latter profits only very little from available a priori knowledge (the slight gains are due to demapping of a multilevel modulation). Note that a SIC without error propagation (GenieSIC) is capacity-achieving, which explains the good performance in the high LA,Det regime. The proposed adaptive SoftSIC/LD detector (circle markers) always performs at least as good as the SoftSIC detector and shows performance close to that of the linear detector in the low LA regime, showing that it realizes the potential gains of switching between the two techniques. IV.

S IMULATION R ESULTS

The proposed algorithms were tested by simulating the home/office setup of the WIGWAM broadband MIMO-OFDM system [10] (160 MHz FFT bandwidth, 1024 point FFT,

Figure 3 shows the performance of the three investigated detection techniques in a non-iterative setup, for the low diversity regime (results for the high diversity case are similar; therefore not shown). The proposed adaptive SoftSIC/LD detector consistently outperforms both the linear and the SoftSIC detector. However, the achieved gains are relatively low (in the order of 0.5dB at a target FER of 1%). The SoftSIC detector’s performance continually degrades as the modulation order increases. It only achieves linear MMSE performance for 16-QAM transmission (for 4-QAM transmission it slightly outperforms the linear detector; results not shown). In light of the effort invested in the calculation of soft symbols and interference cancellation noise for both the SoftSIC and the SoftSIC/LD detector, using a simple linear MMSE detector might be the wisest choice in such a setup. Figure 4 presents results for an iterative setup (4 detector-decoder iterations). Again, the SoftSIC/LD detector performs as good as or better than the other detection techniques (within the limits of precision). The same performance can, however, be achieved by the hybrid detector setup (linear detection in the first, SoftSIC in the 3 subsequent iterations). Since the highest effort for detection is invested for the SoftSIC/LD in the first iteration, and substantial savings are possible for the last 3 iterations [9], this setup appears to be the more attractive option. Both investigated combined linear/SoftSIC receiver techniques achieve performance within 1-2dB of the Sphere detector (1024 candidates, 4 detectordecoder iterations) bound, and achieve a gain of 1.5-2dB over ML detection (Sphere, 1024 candidates, single shot detection-

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)

0

10

256−QAM 16−QAM

−1

FER

10

64−QAM x ... 1 x LD, 3 x SoftSIC + ... 4 x SoftSIC o ... 4 x SoftSIC/LD v ... 1 x Sphere ^ ... 4 x Sphere 9 11 13

−2

10

15

17 SNR [dB]

19

21

23

25

Figure 4: Performance results for iterative equalization, IEEE 802.11n E channel model

which switches between the two detector types, based on the quality of the received signal and available soft information. We compared the performance of this technique with that of simple linear MMSE and SoftSIC detection in a non-iterative setup and showed that the proposed detection algorithm outperforms both schemes. For the case of iterative MIMO equalization, we compared the performance of the proposed scheme with that of a hybrid detector (a linear MMSE us used in the first and a SoftSIC in all subsequent iterations). While the adaptive and the hybrid MIMO detector show essentially the same performance, the latter yields the best performancecomplexity trade-off, as it avoids the complexity of SoftSIC detection in the first receiver iteration. Results for a low diversity environment suggest that high performance is achievable by the investigated low complexity detection techniques mainly when enough frequency (or time) diversity is available to make up for the loss in spatial diversity caused by such simple schemes. ACKNOWLEDGMENT

decoding). Note that by appropriate scaling of the number of internal decoder iterations as a function of the available a priori knowledge [11], the decoding complexity for the iterative setup can be substantially reduced, rendering the combination of linear and SoftSIC detectors are very promising alternative to Sphere detection. The results presented in Figure 5 (channel

16−QAM

256−QAM

[2] B. M. Hochwald and S. ten Brink, “Achieving Near-Capacity on a Multiple-Antenna Channel,” IEEE Transactions on Communications, vol. 51, no. 3, pp. 389–399, Mar. 2003. [3] W. J. Choi, K. W. Cheong, and J. M. Cioffi, “Iterative Soft Interference Cancellation for Multiple Antenna Systems,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC’00), no. 1, 2000, pp. 304–309.

−1

FER

10

64−QAM

−2

R EFERENCES [1] G. Foschini and M. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” IEEE Journal on Wireless Personal Communications, vol. 6, no. 3, pp. 311–335, Oct. 1998.

0

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This work was supported by the German ministry of research and education within the project Wireless Gigabit with advanced multimedia support (WIGWAM) under grant 01 BU 370.

[4] P. Robertson, E. Villebrun, and P. Hoeher, “A comparison of optimal and suboptimal MAP decoding algorithms operating in the log domain,” in Proc. IEEE International Conference on Communications (ICC’95).

x ... 1 x LD, 3 x SoftSIC + ... 4 x SoftSIC o ... 4 x SoftSIC/LD v ... 1 x Sphere ^ ... 4 x Sphere

10

12

14

16

18

20 SNR [dB]

22

24

26

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30

Figure 5: Performance results for iterative equalization, IEEE 802.11n B channel model model B, again 4 detector-decoder iterations) show that the behavior is similar in the low diversity environment. The offset to the Sphere Detector bound is only slightly increasing to 2-3 dB. The availability of a reasonable amount of frequency diversity is hence key to achieving very good performance with low complexity detection techniques in combination with low-rate channel coding. V.

C ONCLUSIONS

In this paper, we studied the performance of linear MMSE and SoftSIC detectors, and combination of the two detector types, for the application in iterative and non-iterative MIMO receivers. We proposed a new adaptive SoftSIC/LD detector

[5] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate,” IEEE Transactions on Information Theory, vol. 20, pp. 248–287, 1974. [6] D. W¨ubben, “Effiziente Detektionsverfahren f¨ur Multilayer-Systeme,” PhD Thesis, Dec. 2005. [7] D. W¨ubben, R. B¨ohnke, V. Kuehn, and K. Kammeyer, “MMSE Extension of V-BLAST based on Sorted QR Decomposition,” in IEEE Semiannual Vehicular Technology Conference (VTC2003-Fall), Orlando, USA, Oct. 2003. [8] B. Hassibi, “An Efficient Square Root Algorithm for BLAST,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’00), vol. 2, Istanbul, Turkey, June 2000, pp. 737 – 740. [9] S. Bittner, E. Zimmermann, and G. Fettweis, “Low Complexity Soft Interference Cancellation for MIMO Systems,” in Proc. IEEE Vehicular Technology Conference (VTC Spring’06), Melbourne, Australia, May 2006. [10] G. Fettweis, “WIGWAM: System Concept for 1GBit/s and Beyond,” in IEEE 802 Plenary Meeting (Tutorial Presentation), Vancouver, Canada, Nov. 2005. [11] E. Zimmermann, S. Bittner, and G. Fettweis, “Complexity Reduction in Iterative MIMO Receivers Based on EXIT Chart Analysis,” in Proc. ISTC/SCC’06, Munich, Germany, Apr. 2006.